Proof of 1+1=2: Maths Explanation & Examples

[nebbione]

Proof of 1+1=2 ?

Hi everyone! Which is the proof of the rightness of arithmetical operations ?
For example which is the proof that 1+1=2 ??
Can you link me the proof or explain how it is done or where i have to start ?

 

[acabus]

It's proven, after hundreds of pages, in this book:
http://en.wikipedia.org/wiki/Principia_mathematica

I definitely can't explain how it's done, it's all in symbolic logic, and I guess you start with a degree level mathematical education.

 

[nebbione]

Ahhh ok ok ! But there isn't a simpler proof ? and who can proof that 1*1 = 1 ?? why it is like that ? which is the proof ?

 

[acabus]

There aren't any proofs if you keep on asking "why?", at some point or another you're going to have to accept things as true, without proof: these are axioms. Some common axioms for arithmetic are:

1) a + (b + c) = (a + b) + c
2) a + b = b + a
3) a + 0 = a
4) a + (-a) = 0
5) a*b = b*a
6) a*(b*c) = (a*b)*c
7) 1*a = a
8) a*(1/a) = 1
9) a*(b+c) = a*b + a*c

There are more formal definitions using set theory. Anyway, the idea is to make the axioms as simple as self-evidently true as possible, and then derive everything else from them.

Also, have you heard of the parallel postulate and non-Euclidean geometry? That's quite interesting.

EDIT: Although, re-reading your post, you may have meant something different with your first question. You could think of 2 as being defined as the answer to 1 + 1.

 

[Whovian]

As the previous poster said, at some point, there are no proofs. Before we can prove 1+1=2, we have to define what addition of two real numbers. And we define that ... how? We just take it as something that "is." Real numbers prove difficult to define, but it turns out that once we have sets defined (sets may be what we have to take for granted,) it's possible to define real numbers. This is part of why some people think we invented mathematics, but that's a thread in the Philosophy forum.

 

[nebbione]

OK nice argument! I understand now! Doing some study in linear algebra i remember about neutral element in multiplication (that is 1) and neutral element in addition (that is 0)
so i can proof that 1*1 is equal to 1 by this axioms right ?
Now it's clear